The present invention relates to power-flow/voltage control in utility/industrial power networks of the types including many power plants/generators interconnected through transmission/distribution lines to other loads and motors. Each of these components of the power network is protected against unhealthy or alternatively faulty, over/under voltage, and/or over loaded damaging operating conditions. Such a protection is automatic and operates without the consent of power network operator, and takes an unhealthy component out of service by disconnecting it from the network. The time domain of operation of the protection is of the order of milliseconds.
The purpose of a utility/industrial power network is to meet the electricity demands of its various consumers 24-hours a day, 7-days a week while maintaining the quality of electricity supply. The quality of electricity supply means the consumer demands be met at specified voltage and frequency levels without over loaded, under/over voltage operation of any of the power network components. The operation of a power network is different at different times due to changing consumer demands and development of any faulty/contingency situation. In other words healthy operating power network is constantly subjected to small and large disturbances. These disturbances could be consumer/operator initiated, or initiated by overload and under/over voltage alleviating functions collectively referred to as security control functions and various optimization functions such as economic operation and minimization of losses, or caused by a fault/contingency incident.
For example, a power network is operating healthy and meeting quality electricity needs of its consumers. A fault occurs on a line or a transformer or a generator which faulty component gets isolated from the rest of the healthy network by virtue of the automatic operation of its protection. Such a disturbance would cause a change in the pattern of power flows in the network, which can cause over loading of one or more of the other components and/or over/under voltage at one or more nodes in the rest of the network. This in turn can isolate one or more other components out of service by virtue of the operation of associated protection, which disturbance can trigger chain reaction disintegrating the power network.
Therefore, the most basic and integral part of all other functions including optimizations in power network operation and control is security control. Security control means controlling power flows so that no component of the network is over loaded and controlling voltages such that there is no over voltage or under voltage at any of the nodes in the network following a disturbance small or large. As is well known, controlling electric power flows include both controlling real power flows which is given in MWs, and controlling reactive power flows which is given in MVARs. Security control functions or alternatively overloads alleviation and over/under voltage alleviation functions can be realized through one or combination of more controls in the network. These involve control of power flow over tie line connecting other utility network, turbine steam/water/gas input control to control real power generated by each generator, load shedding function curtails load demands of consumers, excitation controls reactive power generated by individual generator which essentially controls generator terminal voltage, transformer taps control connected node voltage, switching in/out in capacitor/reactor banks controls reactive power at the connected node.
Control of an electrical power system involving power-flow control and voltage control commonly is performed according to a process shown in FIG. 5. The various steps entail the following.    Step-10: Obtain on-line/simulated readings of open/close status of all switches and circuit breakers, and read data of maximum and minimum reactive power generation capability limits of PV-node generators, maximum and minimum tap positions limits of tap changing transformers, and maximum power carrying capability limits of transmission lines, transformers in the power network, or alternatively read data of operating limits of power network components.    Step-20: Obtain on-line readings of real and reactive power assignments/schedules/specifications/settings at PQ-nodes, real power and voltage magnitude assignments/schedules/specifications/settings at PV-nodes and transformer turns ratios. These assigned/set values are controllable and are called controlled variables/parameters.    Step-30: Resulting voltage magnitudes and angles at power network nodes, power flows through various power network components, reactive power generations by generators and turns ratios of transformers in the power network are determined by performance of loadflow calculation, for the assigned/set/given/known values from step-20 or previous process cycle step-60, of controlled variables/parameters.    Step-40: The results of Loadflow calculation of step-30 are evaluated for any over loaded power network components like transmission lines and transformers, and over/under voltages at different nodes in the power system    Step-50: If the system state is acceptable implying no over loaded transmission lines and transformers and no over/under voltages, the process branches to step-70, and if otherwise, then to step-60    Step-60: Changes the controlled variables/parameters set in step-20 or as later set by the previous process cycle step-60 and returns to step-30    Step-70: Actually implements the corrected controlled variables/parameters to obtain secure/correct/acceptable operation of power system
Overload and under/over voltage alleviation functions produce changes in controlled variables/parameters in step-60 of FIG. 5. In other words controlled variables/parameters are assigned or changed to the new values in step-60. This correction in controlled variables/parameters could be even optimized in case of simulation of all possible imaginable disturbances including outage of a line and loss of generation for corrective action stored and made readily available for acting upon in case the simulated disturbance actually occurs in the power network. In fact simulation of all possible imaginable disturbances is the modern practice because corrective actions need be taken before the operation of individual protection of the power network components.
It is obvious that loadflow calculation consequently is performed many times in real-time operation and control environment and, therefore, efficient and high-speed loadflow calculation is necessary to provide corrective control in the changing power system conditions including an outage or failure of any of the power network components. Moreover, the loadflow calculation must be highly reliable to yield converged solution under a wide range of system operating conditions and network parameters. Failure to yield converged loadflow solution creates blind spot as to what exactly could be happening in the network leading to potentially damaging operational and control decisions/actions in capital-intensive power utilities.
The power system control process shown in FIG. 5 is very general and elaborate. It includes control of power-flows through network components and voltage control at network nodes. However, the control of voltage magnitude at connected nodes within reactive power generation capabilities of electrical machines including generators, synchronous motors, and capacitor/inductor banks, and within operating ranges of transformer taps is normally integral part of load flow calculation as described in “LTC Transformers and MVAR violations in the Fast Decoupled Load Flow, IEEE Trans., PAS-101, No. 9, PP. 3328-3332, September 1982.” If under/over voltage still exists in the results of load flow calculation, other control actions, manual or automatic, may be taken in step-60 in the above and in FIG. 5. For example, under voltage can be alleviated by shedding some of the load connected.
The prior art and present invention are described using the following symbols and terms:     Ypq=Gpq+jBpq: (p-q)th element of nodal admittance matrix without shunts     Ypp=Gpp+jBpp: p-th diagonal element of nodal admittance matrix without shunts     yp=gp+jbp: total shunt admittance at any node-p     Vp=ep+jfp=Vp∠θp: complex voltage of any node-p    Δθp, ΔVp: voltage angle, magnitude corrections    Δep, Δfp: real, imaginary components of voltage corrections    Pp+jQp: net nodal injected power calculated    ΔPp+jΔQp: nodal power residue or mismatch    RPp+jRQp: modified nodal power residue or mismatch    Φp: rotation or transformation angle    [RP]: vector of modified real power residue or mismatch    [RQ]: vector of modified reactive power residue or mismatch    [Yθ]: gain matrix of the P-θ loadflow sub-problem defined by eqn. (1)    [YV]: gain matrix of the Q-V loadflow sub-problem defined by eqn. (2)    m: number of PQ-nodes    k: number of PV-nodes    n=m+k+1: total number of nodes    q>p: q is the node adjacent to node-p excluding the case of q=p    [ ]: indicates enclosed variable symbol to be a vector or a matrix    LRA: Limiting Rotation Angle, −36° for prior art, −48° for invented models    PQ-node: load-node, where, Real-Power-P and Reactive-Power-Q are specified    PV-node: generator-node, where, Real-Power-P and Voltage-Magnitude-V are specified     Ypq′=Gpq′+jBpq′: rotated (p-q)th element of nodal admittance matrix without shunts     Ypp′=Gpp′+jBpp′: rotated p-th diagonal element of nodal admittance matrix without shunts    ΔPp′=ΔPp Cos Φp+ΔQp Sin Φp: rotated or transformed real power mismatch    ΔQp′=ΔQp Cos Φp−ΔPp Sin Φp: rotated or transformed reactive power mismatch    Loadflow Calculation: Each node in a power network is associated with four electrical quantities, which are voltage magnitude, voltage angle, real power, and reactive power. The loadflow calculation involves calculation/determination of two unknown electrical quantities for other two given/specified/scheduled/set/known electrical quantities for each node. In other words the loadflow calculation involves determination of unknown quantities in dependence on the given/specified/scheduled/set/known electrical quantities.    Loadflow Model: a set of equations describing the physical power network and its operation for the purpose of loadflow calculation. The term ‘loadflow model’ can be alternatively referred to as ‘model of the power network for loadflow calculation’. The process of writing Mathematical equations that describe physical power network and its operation is called Mathematical Modeling. If the equations do not describe/represent the power network and its operation accurately the model is inaccurate, and the iterative loadflow calculation method could be slow and unreliable in yielding converged loadflow calculation. There could be variety of Loadflow Models depending on organization of set of equations describing the physical power network and its operation, including Decoupled Loadflow Models, Super Decoupled Loadflow Models, Fast Super Decoupled Loadflow (FSDL) Model, and Super Super Decoupled Loadflow (SSDL) Model.    Loadflow Method: sequence of steps used to solve a set of equations describing the physical power network and its operation for the purpose of loadflow calculation is called Loadflow Method, which term can alternatively be referred to as ‘loadflow calculation method’ or ‘method of loadflow calculation’. One word for a set of equations describing the physical power network and its operation is: Model. In other words, sequence of steps used to solve a Loadflow Model is a Loadflow Method. The loadflow method involves definition/formation of a loadflow model and its solution. There could be variety of Loadflow Methods depending on a loadflow model and iterative scheme used to solve the model including Decoupled Loadflow Methods, Super Decoupled Loadflow Methods, Fast Super Decoupled Loadflow (FSDL) Method, and Super Super Decoupled Loadflow (SSDL) Method. All decoupled loadflow methods described in this application use either successive (1θ, 1V) iteration scheme or simultaneous (1V, 1θ) iteration scheme, defined in the following.
Prior art method of loadflow calculation of the kind carried out as step-30 in FIG. 5, include a class of methods known as decoupled loadflow. This class of methods consists of decoupled loadflow and super decoupled loadflow methods including Fast Super Decoupled Loadflow method. However, functional forms of different elements of the prior art Super Decoupled Loadflow Fast Super Decoupled Loadflow (FSDL) model defined by system of equations (1) and (2) will be given below before description of steps of the prior art loadflow calculation method. The prior art FSDL model is very sensitive, in terms of iterations required to achieve converged loadflow calculation, to the restricted value of the rotation angle applied to complex power mismatch in terms of mismatch in real and reactive power flowing in through each of PQ-nodes. Moreover, the presence of twice the value of transformed network shunt in a diagonal element of the gain matrix [YV] for certain power network causes slow convergence taking increased number of iterations to converge to a solution, and therefore required increased calculation time.
The aforesaid class of Decoupled Loadflow models involves a system of equations for the separate calculation of voltage angle and voltage magnitude corrections. Each decoupled model comprises a system of equations (1) and (2) differing in the definition of elements of [RP], [RQ], [Yθ] and [YV].[RP]=[Yθ][Δθ]  (1)[RQ]=[YV][ΔV]  (2)
A decoupled loadflow calculation method involves solution of a decoupled loadflow model comprising system of equations (1) and (2) in an iterative manner. Commonly, successive (1θ, 1V) iteration scheme is used for solving system of equations (1) and (2) alternately with intermediate updating. Each iteration involves one calculation of [RP] and [Δθ] to update [θ] and then one calculation of [RQ] and [ΔV] to update [V]. The sequence of equations (3) to (6) depicts the scheme.[Δθ]=[Yθ]−1[RP]  (3)[θ]=[θ]+[Δθ]  (4)[ΔV]=[YV]−1[RQ]  (5)[V]=[V]+[ΔV]  (6)
The elements of [RP] and [RQ] for PQ-nodes are given by equations (7) to (10).RPp=(ΔPp Cos Φp+ΔQp Sin Φp)Vp=ΔPp′/Vp  (7)RQp=(−ΔPp Sin Φp+ΔQp Cos Φp)/Vp=ΔQp′/Vp  (8)Cos Φp=Absolute(Bpp/√{square root over ((Gpp2+Bpp2))})≧Cos(−36°)  (9)Sin Φp=−Absolute(Gpp/√{square root over ((Gpp2+Bpp2))})≧Sin(−36°)  (10)
A description of Super Decoupling principle and the prior art FSDL model is given in, “Fast Super Decoupled Loadflow”, IEE proceedings Part-C, Vol. 139, No. 1, pp. 13-20, January 1992.
Fast Super Decoupled Loadflow (FSDL) model consists of equations (3) to (16).RPp=ΔPp/(KpVp)  (11)
                              Y          ⁢                                          ⁢                      θ            pq                          =                  [                                                                                                                             -                                        ⁢                                          Y                      pq                                                                                                                                                          -                                            ⁢                      for                      ⁢                                                                                          ⁢                      branch                      ⁢                                                                                          ⁢                      r                      ⁢                                              /                                            ⁢                      x                      ⁢                                                                                          ⁢                      ratio                                        ≤                    2.0                                                                                                                                          -                                        ⁢                                          (                                                                        B                          pq                                                +                                                  0.9                          ⁢                                                      (                                                                                          Y                                pq                                                            -                                                              B                                pq                                                                                      )                                                                                              )                                                                                                                                                          -                                            ⁢                      for                      ⁢                                                                                          ⁢                      branch                      ⁢                                                                                          ⁢                      r                      ⁢                                              /                                            ⁢                      x                      ⁢                                                                                          ⁢                      ratio                                        >                    2.0                                                                                                                                          -                                        ⁢                                          B                      pq                                                                                                                                                                                                                                              -                                                        ⁢                            for                            ⁢                                                                                                                  ⁢                            branches                            ⁢                                                                                                                  ⁢                            connected                            ⁢                                                                                                                  ⁢                            between                            ⁢                                                                                                                  ⁢                            two                            ⁢                                                                                                                  ⁢                            PV                            ⁢                                                          -                                                        ⁢                            nodes                                                                                                                                                                            or                            ⁢                                                                                                                  ⁢                            a                            ⁢                                                                                                                  ⁢                            PV                            ⁢                                                          -                                                        ⁢                            node                            ⁢                                                                                                                                                      ⁢                                                                                                                          ⁢                                                                                                                                                    ⁢                            and                            ⁢                                                                                                                  ⁢                            the                            ⁢                                                                                                                  ⁢                            slack                            ⁢                                                          -                                                        ⁢                            node                                                                                                                ⁢                                                                                                                                                                        (        12        )                                                          ⁢                              YV            pq                    =                      [                                                                                                      -                                        ⁢                                          Y                      pq                                                                                                                                                          -                                            ⁢                      for                      ⁢                                                                                          ⁢                      branch                      ⁢                                                                                          ⁢                      r                      ⁢                                              /                                            ⁢                      x                      ⁢                                                                                          ⁢                      ratio                                        ≤                    2.0                                                                                                                                          -                                        ⁢                                          (                                                                        B                          pq                                                +                                                  0.9                          ⁢                                                      (                                                                                          Y                                pq                                                            -                                                              B                                pq                                                                                      )                                                                                              )                                                                                                                                                          -                                            ⁢                      for                      ⁢                                                                                          ⁢                      branch                      ⁢                                                                                          ⁢                      r                      ⁢                                              /                                            ⁢                      x                      ⁢                                                                                          ⁢                      ratio                                        >                    2.0                                                                                                          (        13        )                                                          ⁢                              Y            ⁢                                                  ⁢                          θ              pp                                =                                                                                          ∑                                          q                      >                      p                                                        ⁢                                                            -                      Y                                        ⁢                                                                                  ⁢                                          θ                      pq                                                                                                  and                                                                                  YV                    pp                                    =                                                                                    -                                            ⁢                      2                      ⁢                                                                                          ⁢                                              b                        p                        ′                                                              +                                                                  ∑                                                  q                          >                          p                                                                    ⁢                                              -                                                  YV                          pq                                                                                                                                                                            (        14        )            bp′=bp Cos Φp or bp′=bp  (15)Kp=Absolute(Bpp/Yθpp)  (16)
Branch admittance magnitude in (12) and (13) is of the same algebraic sign as its susceptance. Elements of the two gain matrices differ in that diagonal elements of [YV] additionally contain the b′ values given by equation (15) and in respect of elements corresponding to branches connected between two PV-nodes or a PV-node and the slack-node. In two simple variations of the FSDL model, one is to make YVpq=Yθpq and the other is to make Yθpq=YVpq.
The steps of loadflow calculation method FSDL are shown in the flowchart of FIG. 1. Referring to the flowchart of FIG. 1, different steps are elaborated in steps marked with similar letters in the following. The words “Read system data” in Step-a correspond to step-10 and step-20 in FIG. 5, and step-14, step-20, step-32, step-44, step-50 in FIG. 6. All other steps in the following correspond to step-30 in FIG. 5, and step-60, step-62, and step-64 in FIG. 6.    a. Read system data and assign an initial approximate solution. If better solution estimate is not available, set all the nodes voltage magnitudes and angles equal to those of the slack-node. This is referred to as the slack-start.    b. Form nodal admittance matrix, and Initialize iteration count ITRP=ITRQ=r=0.    c. Compute sine and cosine of rotation angles using equations (9) and (10), and store them. If Cos Φp<Cos(−36°), set Cos Φp=Cos(−36°) and Sin Φp=Sin(−36°).    d. Form (m+k)×(m+k) size gain matrices [Yθ] and [YV] of (1) and (2) respectively each in a compact storage exploiting sparsity, using equations (12) to (15). In [YV] matrix, replace diagonal elements corresponding to PV-nodes by very large value, say, 10.010. Factorize [Yθ] and [YV] using the same ordering of nodes regardless of node-types and store them using the same indexing and addressing information.    e. Compute residues [ΔP] at PQ- and PV-nodes and [ΔQ] at only PQ-nodes. If all are less than the tolerance (ε), proceed to step-m. Otherwise follow the next step.    f. Compute the vector of modified residues [RP] using (7) for PQ-nodes, and using (11) and (16) for PV-nodes.    g. Solve (3) for [Δθ] and update voltage angles using, [θ]=[θ]+[Δθ].    h. Set voltage magnitudes of PV-nodes equal to the specified values, and Increment the iteration count ITRP=ITRP+1 and r=(ITRP+ITRQ)/2.    i. Compute residues [ΔP] at PQ- and PV-nodes and [ΔQ] at PQ-nodes only. If all are less than the tolerance (ε), proceed to step-m. Otherwise follow the next step.    j. Compute the vector of modified residues [RQ] using (8) for only PQ-nodes.    k. Solve (5) for [ΔV] and update PQ-node magnitudes using [V]=[V]+[ΔV]. While solving equation (5), skip all the rows and columns corresponding to PV-nodes.    l. Calculate reactive power generation at PV-nodes and tap positions of tap-changing transformers. If the maximum and minimum reactive power generation capability and transformer tap position limits are violated, implement the violated physical limits and adjust the loadflow solution by the method described in “LTC Transformers and MVAR violations in the Fast Decoupled Load Flow, IEEE Trans., PAS-101, No. 9, PP. 3328-3332, September 1982”.    m. Increment the iteration count ITRQ=ITRQ+1 and r=(ITRP+ITRQ)/2, and Proceed to step-d.    n. From calculated values of voltage magnitude and voltage angle at PQ-nodes, voltage angle and reactive power generation at PV-nodes, and tap position of tap changing transformers, calculate power flows through power network components.
In super decoupled loadflow models [Yθ] and [YV] are real, sparse, symmetrical and built only from network elements. Since they are constant, they need to be factorized once only at the start of the solution. Equations (1) and (2) are to be solved repeatedly by forward and backward substitutions. [Yθ] and [YV] are of the same dimensions (m+k)×(m+k) when only a row/column of the slack-node or reference-node is excluded and both are triangularized using the same ordering regardless of the node-types. For a row/column corresponding to a PV-node excluded in [YV], use a large diagonal to mask out the effects of the off-diagonal terms. When the PV-node is switched to the PQ-type, removing the large diagonal reactivates the row/column corresponding to a switched PV-node to PQ-node type. This technique is especially useful in the treatment of PV-nodes in the gain matrix [YV].
The convergence of the prior art FSDL method is very sensitive to the value of the restriction applied to the rotation angle. The best possible convergence from non-linearity consideration could be achieved by restricting rotation angle to maximum of −36. However, when large Resistance (R)/Reactance (X) ratio branch is present in the network without creating non-linearity problem, it takes large number of iterations to converge. Moreover, the presence of twice the transformed value of network shunts in the diagonal elements of the gain matrix [YV], causes it to take increased number of iterations in case of certain power networks. These problems are overcome by formulating power flow equations such that transformed values of known/given/specified/scheduled/set quantities appears in the diagonal elements of the gain matrix [YV] as described in the following.